Uniqueness of bounded semigeostrophic potential vorticity
Let $\Omega \subset \mathbb{R}^3$ be a ball of volume one and set $J(a, b, c) = (-b, a, 0)$. For a compactly supported probability density $\rho$ on $\mathbb{R}^3$ let $T[\rho] = \nabla P^*$ be the optimal transport map, for the quadratic cost, carrying $\rho(y)\,dy$ to $\mathbf{1}_\Omega(x)\,dx$. In geostrophic coordinates the incompressible semigeostrophic equations are the active transport equation $$\partial_t \rho + \nabla_y \cdot \bigl[ \rho \, J\bigl(y - T[\rho_t](y)\bigr) \bigr] = 0$$ for the potential vorticity $\rho_t$. Global weak solutions exist for initial densities in $L^p$, $p > 3$, with compact support (Benamou and Brenier), for $p \ge 1$ (Lopes Filho and Nussenzveig Lopes), and for compactly supported initial measures (Loeper; Feldman and Tudorascu); Bourne, Egan, Pelloni and Wilkinson gave a constructive proof by semi-discrete optimal transport. On uniqueness, Loeper proved local-in-time uniqueness of Hölder continuous periodic solutions, and Feldman and Tudorascu weak–strong uniqueness under uniform convexity.
Problem. Fix a compactly supported probability density $\rho_0 \in L^\infty(\mathbb{R}^3)$. Is there at most one distributional solution of this equation with $\rho|_{t=0} = \rho_0$ among families of nonnegative probability densities $\rho_t$ that are narrowly continuous in time and, on each finite interval $[0, T]$, uniformly bounded in $L^\infty$ and supported in a common compact set? Narrow continuity means continuity of $t \mapsto \int \phi \rho_t$ for every bounded continuous $\phi$; two solutions are equal if their measures agree at every time.
Source: The uniqueness question for weak solutions of the semigeostrophic equations in geostrophic coordinates, noted as open by Bourne, Egan, Pelloni and Wilkinson (2022), §1, and Lavier (2025), Remark 1.1; stated by AIM — Open Applied Problems for bounded densities, with a ball as the physical domain.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, optimal transport, semigeostrophic equations, weak solutions
References
@article{bourne2022,
author = {Bourne, David P. and Egan, Charlie P. and Pelloni, Beatrice and Wilkinson, Mark},
title = {Semi-discrete optimal transport methods for the semi-geostrophic equations},
journal = {Calc. Var. Partial Differential Equations},
volume = {61},
number = {1},
year = {2022},
pages = {39},
doi = {10.1007/s00526-021-02133-z},
eprint = {2009.04430},
archivePrefix = {arXiv},
}
@article{lavier2025,
author = {Lavier, Th{\'e}o},
title = {A semi-discrete optimal transport scheme for the {3D} incompressible semi-geostrophic equations},
journal = {IMA J. Appl. Math.},
volume = {90},
number = {5},
year = {2025},
pages = {443--464},
doi = {10.1093/imamat/hxaf023},
eprint = {2411.00575},
archivePrefix = {arXiv},
}
@article{benamou1998,
author = {Benamou, Jean-David and Brenier, Yann},
title = {Weak existence for the semigeostrophic equations formulated as a coupled {M}onge--{A}mp{\`e}re/transport problem},
journal = {SIAM J. Appl. Math.},
volume = {58},
number = {5},
year = {1998},
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}
@article{lopesfilho2002,
author = {Lopes Filho, Milton C. and Nussenzveig Lopes, Helena J.},
title = {Existence of a weak solution for the semigeostrophic equation with integrable initial data},
journal = {Proc. Roy. Soc. Edinburgh Sect. A},
volume = {132},
number = {2},
year = {2002},
pages = {329--339},
doi = {10.1017/S0308210500001657},
}
@article{loeper2006,
author = {Loeper, Gr{\'e}goire},
title = {A fully nonlinear version of the incompressible {E}uler equations: the semigeostrophic system},
journal = {SIAM J. Math. Anal.},
volume = {38},
number = {3},
year = {2006},
pages = {795--823},
doi = {10.1137/050629070},
eprint = {math/0504138},
archivePrefix = {arXiv},
}
@article{feldman2015,
author = {Feldman, Mikhail and Tudorascu, Adrian},
title = {On the semi-geostrophic system in physical space with general initial data},
journal = {Arch. Ration. Mech. Anal.},
volume = {218},
number = {1},
year = {2015},
pages = {527--551},
doi = {10.1007/s00205-015-0865-y},
}
@article{feldman2017,
author = {Feldman, Mikhail and Tudorascu, Adrian},
title = {The semi-geostrophic system: weak-strong uniqueness under uniform convexity},
journal = {Calc. Var. Partial Differential Equations},
volume = {56},
number = {6},
year = {2017},
pages = {158},
doi = {10.1007/s00526-017-1254-1},
}
@misc{aim-semigeostrophic-bounded-vorticity-uniqueness,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Uniqueness of bounded semigeostrophic potential vorticity}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/389-semigeostrophic-bounded-vorticity-uniqueness.md},
note = {Entry 389 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00896, "Uniqueness of bounded semigeostrophic potential vorticity", clopen.solutions.